Question

Derive an expression for the resistivity of a conductor in terms of number density of free electrons and relaxation time

Answer 1

The derivation of an expression for the resistivity of a conductor in terms of number density of free electrons and relaxation time is as follows:

• The drift velocity of electrons is given by,
• vd = eE/m Τ
• where, Τ = relaxation time
• electric field, E = V/I
• so, we get,
• vd = eV/mI Τ
• we also know that,
• I = Anevd
• Therefore,
• I = Ane [eV/mI Τ]
• I = [Ane²/mI Τ] V
• we know that,
• R = V/I
• so, we have,
• I/V = [Ane²/mI Τ]
• ⇒ R = V/I = [ml/Ane²Τ]  ......(1)
• The standard formula for resistance is given by,
• R = I/ρ A  .....(2)
• comparing equations (1) and (2), we get
• ml/Ane²Τ = I/ρ A
• ρ = m/ne²Τ
• Hence the required expression.

Answer 2

Answer:

ρ = m/ne²Τ

Explanation:

The drift velocity of electrons is given by,

vd = eE/m Τ

where, Τ = relaxation time

electric field, E = V/I

so, we get,

vd = eV/mI Τ

we also know that,

I = Anevd

Therefore,

I = Ane [eV/mI Τ]

I = [Ane²/mI Τ] V

we know that,

R = V/I

so, we have,

I/V = [Ane²/mI Τ]

⇒ R = V/I = [ml/Ane²Τ]  ......(1)

The standard formula for resistance is given by,

R = I/ρ A  .....(2)

comparing equations (1) and (2), we get

ml/Ane²Τ = I/ρ A

ρ = m/ne²Τ

Hence the required expression.

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